The area of a rhombus is 84 square centimetres and the length of one of its diagonals is 7 centimetres. Find the length, in centimetres, of the other diagonal of the rhombus.

Introduction / Context:
This question tests the relationship between the area of a rhombus and its diagonals. Rhombus based questions are common in aptitude tests because they combine geometry formulas with simple algebra. If you know how the diagonals of a rhombus determine its area, you can easily find an unknown diagonal when the area and one diagonal are given.

Given Data / Assumptions:

• Area of the rhombus = 84 square centimetres.
• Length of one diagonal = 7 centimetres.
• The other diagonal is unknown and must be found.
• The figure is a standard rhombus, so the usual formula for area applies.

Concept / Approach:
The area of a rhombus can be expressed in terms of its diagonals. If d1 and d2 are the lengths of the diagonals, then the area is given by:
Area = (d1 * d2) / 2.
Here, one diagonal and the area are known, so we rearrange the formula to solve for the second diagonal. This turns the problem into a straightforward algebraic equation in one variable.

Step-by-Step Solution:
Let d1 = 7 cm and let d2 be the unknown diagonal in centimetres. Area = (d1 * d2) / 2. So 84 = (7 * d2) / 2. Multiply both sides by 2 to clear the denominator: 84 * 2 = 7 * d2. 168 = 7 * d2. Now divide both sides by 7: d2 = 168 / 7. d2 = 24 centimetres.

Verification / Alternative check:
We can substitute d1 = 7 and d2 = 24 back into the area formula to verify the result. Area = (7 * 24) / 2. First compute 7 * 24 = 168. Then divide by 2 to get 84. This matches the given area, so the computed value of the second diagonal is correct and consistent with the formula.

Why Other Options Are Wrong:
Option A (12): If d2 = 12, area would be (7 * 12) / 2 = 42 square centimetres, which is only half of the required area. Option C (48): If d2 = 48, area would be (7 * 48) / 2 = 168 square centimetres, which is exactly double the required area. Option D (36): If d2 = 36, area would be (7 * 36) / 2 = 126 square centimetres, which is greater than the given area. Option E (18): If d2 = 18, area would be (7 * 18) / 2 = 63 square centimetres, which is less than 84.

Common Pitfalls:
Many learners mistakenly use the formula for the area of a rectangle or for base times height instead of the diagonal based formula for a rhombus. Another common error is to forget the division by 2 in the formula Area = (d1 * d2) / 2. Some students also reverse operations when solving the equation and may multiply instead of dividing, which leads to incorrect diagonal lengths. Writing the formula clearly and performing algebraic steps slowly helps avoid these mistakes.

Final Answer:
The length of the other diagonal of the rhombus is 24 centimetres.